Girl named Liv go with last name?!

Husband and I both love the name LivCharlotte (nn Livy, LC, Charlie, etc)
but our last name is UNDERWOOD and we are wondering if there is a possibility for teasing!
What do you think? Is this name a dead end?

I've always wanted to use Liv for one of my future kids but the issue is, no matter what surname you have it can sound a bit odd because it's such a common word in the English language on it's own, something we use all the time. May I suggest using Livia and calling her Liv or Livy? That was my intention should I ever use the name. It has nice Ancient Roman connotations too LiviaCharlotte would be beautiful!